Theorem dfon2lem1 36760

Description: Lemma for dfon2 36769. (Contributed by Scott Fenton, 28-Feb-2011.) (Proof shortened by SN, 5-Oct-2026.)

Assertion

Ref	Expression
dfon2lem1	⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}

Proof of Theorem dfon2lem1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		truni 5234	. 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}Tr 𝑦 → Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)})
2		simp2 1155	. . . . 5 ⊢ ((𝜑 ∧ Tr 𝑥 ∧ 𝜓) → Tr 𝑥)
3	2	ss2abi 3994	. . . 4 ⊢ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} ⊆ {𝑥 ∣ Tr 𝑥}
4	3	sseli 3909	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → 𝑦 ∈ {𝑥 ∣ Tr 𝑥})
5		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
6		treq 5225	. . . 4 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
7	5, 6	elab 3636	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ Tr 𝑥} ↔ Tr 𝑦)
8	4, 7	sylib 221	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → Tr 𝑦)
9	1, 8	mprg 3086	1 ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}

This proof depends on syntax axioms:   ∧ w3a 1103   ∈ wcel 2147  {cab 2741  ∪ cuni 4872  Tr wtr 5218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-uni 4873  df-iun 4958  df-tr 5219

This theorem is used by:  dfon2lem3  36762  dfon2lem7  36766